## Abstract

Modeling is an important approach for describing the physical and mechanical properties of asphalt, which can be employed to analyze the properties and optimize the performance of asphalt-based materials. The objective of this review is to elucidate the existing modeling techniques from different scales and summarize the prospects and application of different modeling methods. The continuum model and its practical application at the macroscale are introduced first. However, the macroscopic model cannot explain the roles of the different components in asphalt during the degradation process. The microstructural model at the mesoscale is adopted in order to describe the physical properties of the asphalt mixture, and different numerical methods are presented in order to analyze the microstructural models. Due to the heterogeneous nature of asphalt at the nanoscale, various experimental tools have been used to determine the molecular structures of asphalt. The average model and multi-component model have been proposed in the molecular dynamics simulations. This paper provides a clear understanding of the strength and limitations of each modeling technology at different scales and the applications of multiscale modeling for asphalt materials. The possible challenges of asphalt modeling have been addressed in order to encourage the development of asphalt pavement with higher durability.

## Introduction

Asphalt is a highly complex material that is mainly a byproduct of crude oil distillation. It has been widely used in pavement construction during the rapid urbanization of the 21st century [1]. The physical and chemical properties of asphalt are highly dependent on the classes and sizes of the inner components and structures, which can be characterized by macro, meso, and nano scales [2]. In the last few decades, many research studies have been carried out to investigate the physical and chemical properties of asphalt for the improvement of the overall performance of asphalt pavements [3, 4]. Research work related to asphalt materials has varied in length of scale, because of asphalt’s homogeneous nature at the macroscale but heterogeneous properties at smaller scales. Asphalt is characterized as a multi-phase material, and its thermomechanical performance behaves differently at different scales. As shown in Fig. 1, the research progress for asphalt can be traced from the macroscale to the nanoscale. At the macroscale, the surface course of asphalt pavement is built by the asphalt mixture, which consists of graded aggregates enveloped by asphalt binder and a small amount of air. Compared to stiff and hard aggregates, asphalt binder is soft and more susceptible to temperature changes. The thermal susceptibility of asphalt binder results in conventional asphalt pavements becoming hard and brittle in cold conditions and soft and fluid in hot conditions. Problems such as cracking, rutting, and raveling are common due to the hard-soft transition, especially for aged asphalt pavement that has provided several years of service. The reason for this is that asphalt aging is often accompanied by loss of volatiles and oxidation reactions due to the introduction of polar groups, the increase of the total volume of colloidal agglomerates, and the decrease in viscoelasticity properties [5, 6]. The mechanical deformation, distresses, and aging of asphalt pavement are classified as multiscale phenomena that occur over many different length scales and time scales. However, most of the experimental detection or continuum theories for asphalt are based on the macro perspective and so cannot capture the intrinsic mechanisms of the deterioration of asphalt [7]. The performance of asphalt at the macroscale is determined by its inner structure and explained by its micro-mechanisms. From the atomistic perspective, asphalt consists of millions of different organic molecules that vary widely in polarity and molecular weight. The physical, chemical, and mechanical properties of asphalt are governed by a balance of various molecules of different polarities and sizes. The combination of macro and atomistic research can provide new insights and explanations for the failure mechanism and the proposal of novel modification approaches. The key functional parameters for asphalt include rheological properties such as viscosity and diffusion, thermal properties such as glass transition temperature (*T*_{g}) and thermal conduction coefficient, and mechanical properties such as complex modulus, Young’s moduli, and shear moduli. These functional parameters significantly influence the performance of asphalt and are dependent on accurate modeling approaches for asphalt at different scales. Hence, it is important to understand the modeling technologies of asphalt from a multiscale perspective and to achieve better characterization for the properties of asphalt materials.

Modeling is a critical step in understanding the physicochemical properties of asphalt materials. From sub-atomistic scale to macroscale, a large number of modeling methods and computational techniques have been proposed for exploring the behaviors of asphalt materials and their applications. Modeling techniques and tools can be applied to different length scales and time scales, as shown in Fig. 2. For each modeling scale, the computational model can be characterized by a certain number of atoms. Theoretical and simulation tools include quantum mechanics/density functional theory (DFT), reactive forcefield (ReaxFF) based molecular dynamic (MD) simulations, classic MD simulations, coarse-grained modeling on mesoscale, and continuum mechanics methods. Furthermore, various experimental tools can be applied to detect the behaviors of asphalt materials on different scales. Fourier transform infrared spectroscopy (FTIR), nuclear magnetic resonance (NMR), gel permeation chromatography (GPC), and X-ray diffraction (XRD) have been extensively used for detecting and analyzing asphalt binders [8, 9]. The analyzed information includes the types and percentages of elemental constituents, molecular structures, and polarity behaviors. FTIR spectroscopy is a suitable method for characterizing the chemical composition of asphalt, which is based on the interaction between infrared radiation and the molecules in the sample. The asphalt molecules absorb parts of the infrared radiation and are stimulated to vibrate, and the changing history of functional groups can be obtained by determining the various functional groups in asphalt molecules [10]. Through quantifying and analyzing the changes in the chemical bonds, polymeric additives can also be identified and the interaction between additives and asphalt can be recognized. NMR can be used to detect phenomena at different length scales, which is helpful for investigating the structural characterization aspects of asphalt compositions, such as the percentages of aromatic, methyl carbons, ring carbons, and aromatic carbon ratio. GPC is used to analyze polymers and their homologs, which have the same chemical properties but different molecular volumes. GPC can separate mixtures according to the molecular sizes of asphalt components, and the aging performance of the asphalt binder can be predicted by the chromatograms. Different microscopic techniques, including fluorescence microscopy (FM), scanning electron microscopy (SEM), environmental scanning electron microscopy (ESEM), transmission electron microscopy (TEM), and atomic force microscopy (AFM) have been extensively used to investigate the morphology and microstructure of asphalt [11]. FM has been applied to record the dynamic microcracks and healing process of asphalt and SBS-modified asphalt [12]. SEM is used to characterize asphalt mixture and its profile as well as damage morphology. The molecular structure of the organic reagent, meanwhile, can be recognized using X-ray diffraction. AFM is a scanning probe microscopy with high resolution that can reveal morphology and compare mechanical properties. The surface of the sample is probed by a laser-tracked cantilever with a sharp AFM tip and the force between the tip and sample surface is measured [13]. The correction between SARS fractions and “bee-like structures” has been investigated by AFM, wherein asphaltenes are the main factor in generating this “bee” structure and the alkane plays an important role in prompting the special structures [14]. Through microscopic morphological and mechanical measurement, it has been found that aging and rejuvenation lead to morphological changes [15]. The combination of different detection methods has been applied together for a comprehensive exploration of the properties of asphalt composites. The basic physical properties and microstructural characteristics of asphalt composites have been investigated by a combination of XRD, SEM, FTIR, and GPC [16, 17]. It is noteworthy that the appropriate experimental techniques are highly dependent on the measuring scale and the desired information, as well as the availability and familiarity of each method for researchers. The various techniques and their applications are summarized in Table 1.

Constructing numerical models is the prerequisite step for numerical analyses and simulation of asphalt materials. Research work for asphalt materials begins at the macro level, and the modeling methods are mainly empirical because they arise from the analysis and summary of a large number of experiment results and observations [18, 19]. Classical continuum mechanical theories are the bases for most computational methods applied in civil and mechanical engineering. Ideally, pavement layers are considered as homogeneous, linear elastic, and isotropic under static loads, and the multi-layer elastic theory works well with the layered elastic model. However, the pavement layers are actually heterogeneous and subjected to dynamic and cyclic loadings during normal service. As such, numerical methods are needed to provide better solutions in dynamic analyses of models, taking into consideration the heterogeneity, non-linearity and orthotropy [20, 21]. With the availability of high-speed computers and advanced computer-aided engineering software, numerically computational methods have gained popularity for handling complex geometry, boundary conditions, and material properties. These numerical methods, which include discrete element method (DEM), finite element method (FEM), finite difference method (FDM), and finite volume method (FVM), play a significant role in analyzing the temperature field, structural behaviors, and mechanical response of asphalt pavements [22, 23]. In addition, the extended finite element method (XFEM), which is based on FEM, has been proposed in order to improve the accuracy of solutions for discontinuity problems such as the propagation of cracks, since the meshes near crack tips need to be updated and re-meshed following every step of crack propagation. To avoid grid singularity and distortion of the grid-based or volume-based methods, the element free method (EFM) has a specific ability to solve problems relating to fracture, large deformation, and elastoplasticity [24]. Furthermore, the cohesive zone model (CZM) has been proposed to simulate the cracks of both homogeneous and inhomogeneous materials using the intrinsic cohesive zone, which is an effective method for solving fracture problems in solids [25]. Compared with the macro-mechanical model, the meso-mechanical model of asphalt is more precise, which can provide an insight into the behaviors of asphalt mixture. In the meso-mechanical model, the complex geometry of asphalt pavement can be simplified to its main components, which include aggregates, binder, and air voids. By introducing the meso-mechanical model, an understanding of what really happens inside asphalt pavement and the likely damage to these components can be obtained and analyzed [26]. After incorporating key structural features such as the physical properties, the distributions of aggregates, and the interfacial transmission zone (ITZ), the representative volume element (RVE) for heterogeneous asphalt mixture can be applied in order to obtain the overall thermomechanical response of asphalt materials [27]. However, more computational power is required for calculating and analyzing meso-mechanical models than for macro-level models.

Asphalt materials exhibit unique properties at the atomistic level, where the chemical structures can influence the overall properties of asphalt significantly at the macroscale. It is important to achieve a better understanding of the molecular structures and intermolecular associations of asphalt materials [28]. The emergence of molecular dynamics (MD) simulations not only contributes to the development of bottom-up modeling but also provides inspiration for the design of construction materials [29,30,31,32,33,34,35]. The modeling scheme that uses MD simulations studies the atomistic movements and the evolution of nanostructures in material systems, which can provide an in-depth understanding of the underlying mechanism that governs the macroscopic behaviors of asphalt materials. In MD simulations, materials are treated as thousands of classical particles, which are subjected to Newton’s laws of motion. Three important parameters – displacement, velocity, and acceleration – are calculated as the solution to Newton’s laws in a step-by-step manner. Different numerical methods have been proposed for improving iterative calculation during the main calculation, such as Verlet integration, leapfrog integration, velocity Verlet integration, etc. Moreover, the forcefield plays a critical role in describing the interatomic interaction, and the validity of MD simulations is determined by the selection of forcefield. For MD simulations without any chemical reaction, a non-reactive forcefield can be employed in order to describe the interaction among different components in the material systems. A large number of non-reactive forcefields have been applied in asphalt materials [36,37,38,39]. These classic forcefields have been widely used and proven to be effective in material science. However, the connections among different components can be altered under some special circumstances, meaning that the non-reactive forcefields are no longer accurate. To simulate bond breaking and formation, a bond-order-dependent forcefield, ReaxFF [40], has been developed to describe chemical reactions such as catalysis, pyrolysis, oxidation, and combustion. The parameters for ReaxFF are derived from quantum mechanical (QM) calculations; the accuracy of Reactive MD is close to the density functional theory (DFT) results, but it requires far fewer computational costs than DFT. However, because of the limitation of computational power, MD simulations for asphalt modeling are mainly accessible at nanoscale, which is generally represented as nanoseconds in time scale and nanometers in length scale. The deformation and damage mechanism inside material systems can vary in terms of length scales and time scales, so there is still a need for a robust method that can translate knowledge from the MD simulations to the macroscale models. Predictions for the mechanical behaviors of materials could then be achieved using a bottom-up approach. Moreover, the application of the multiscale modeling method has adapted knowledge from nanoscale to macroscale, and so material behaviors can be analyzed and interpreted from a multiscale perspective [41,42,43,44]. These multiscale modeling methods enable researchers to bridge the gaps in modeling among different scales and allow for predictions of material properties or system behaviors in a comprehensive and integrated aspect.

The objective of this paper is to provide an integrated review of multiscale modeling of asphalt materials from macroscale to nanoscale and demonstrate the development trend in asphalt modeling as well as the transition from research to application. The advantages and limitations of macroscale, mesoscale, and nanoscale modeling for asphalt materials will be introduced and elucidated. For the macroscale and mesoscale, the DEM, FEM, CZM, and RVE models will be introduced and summarized. For the nanoscale, computational models like MD model and the DFT model will be introduced and reviewed. Multiscale modeling technology is significant in terms of improving performance quantitatively at the macroscale, and uncovers the failure mechanism qualitatively at smaller scales. Multiscale modeling also contributes to a thorough and integrated understanding of the properties of asphalt, which is beneficial for the multiscale modification and applications of asphalt materials. Discussion about the prospects and critical challenges of asphalt modeling is presented at the end of this paper.

## Development for Modeling Asphalt Materials

Asphalt has become the most popular material for pavement due to its pleasant rideability and convenient maintenance. However, the performance of asphalt pavement can be seriously damaged by increasing axing loads and harsh environment. It is urgent that we improve the performance of asphalt pavement and prolong its service life. The properties of asphalt binder have been investigated thoroughly, since they can directly influence the performance of asphalt pavement. Asphalt binder becomes soft at high temperatures and rigid at low temperatures, due to its complex chemical and rheological nature. It is important to model the properties of asphalt accurately and elucidate the inner mechanisms of different failure modes. At the macroscale and mesoscale, asphalt is considered a homogeneous binder that wraps the aggregates together. However, problems such as rutting and cracks normally occur at the interface of asphalt and aggregate or inside the asphalt binder, which suggests that asphalt binder is the weaker link inside the asphalt mixture and plays a critical role in the deterioration of asphalt pavement. Therefore, the studies of asphalt have evolved to a much deeper scale. At the nanoscale, asphalt is a heterogeneous material that consists of various hydrocarbon molecules and non-hydrocarbon molecules with different shapes and polarities. It has been found that these asphalt molecules can affect the physical and mechanical properties of asphalt bulk as well as its interaction with the aggregates. Nowadays, research studies on asphalt are developing into different scales, and multiscale modeling is a more promising method for providing insight and analysis of the failure mechanisms of asphalt and exploring its reinforcement.

## Macroscale and Mesoscale Modeling

Many constitutive studies predicting the mechanical behaviors of asphalt materials have been carried out in the last two decades, which can be broadly classified into two main categories: continuum models at the macroscopic level and microstructural models at the mesoscopic level [45]. For continuum models, asphalt materials are studied as a continuum, with the model’s parameters obtained through experimental measurements of the representative samples. The average mechanical responses at the macroscopic level are focused, which is generally also a reflection of the microstructural features. Continuum modeling at the macroscale can be more efficient than microstructural models in predicting the degradations of a practical pavement structure, although it cannot directly elucidate why various constituents in asphalt mixture behave differently under external loadings. It is difficult for continuum modeling to explain changes in the internal structures and explore the reinforcing details of reinforcements. The mechanical behaviors of asphalt materials are strongly correlated to the local load transferring among the wrapped aggregate particles, which is taken as the microstructural response. For microstructural models, the constitutive relations are constructed from the microscopic components of materials and their individual and interactive behaviors. Building microstructural models of asphalt materials costs more than continuum models due to their better geometric and physical characterization. Nevertheless, microstructural models can truly represent the heterogeneous properties of asphalt and achieve higher accuracy than continuum models. The computational microstructural modeling method has become increasingly popular in the asphalt community, since it can separately account for numerical damage modes by considering different individual mixture constituents as well as mixture heterogeneity [46]. Asphalt modeling with microstructures can reduce or even eliminate the experimental cost and result in tremendous potential benefits. Microstructural models allow a thorough examination of microstructural material behaviors, such as strain distribution inside the aggregate skeleton and asphalt matrix, and they can provide a powerful tool for optimizing the mixture design on the basis of mechanical performance.

In a microstructural model, the asphalt mixture is regarded as a multi-phase material, which includes coarse aggregates, air voids, and asphalt mastic containing fine aggregates and asphalt binder. Coarse aggregates are granular and discontinuous, while the asphalt mastic is characterized as a continuum and a viscoelastic medium [47]. The microstructural attributes of aggregates, matrix, and interfacial interactions play a decisive role in the macro-mechanical performance of the asphalt mixture. The important microstructural properties of aggregates include mineralogy, gradation, size, shape, texture, modulus, and packing geometry. Properties such as microcracking, void percentages, viscoelastic responses, and bonding strength are critical in the asphalt matrix [48]. For the matrix/aggregate interface, the important microstructural attributes include adhesion, adsorption, and physicochemical interactions. The strength of the asphalt mixture highly relies on an interlocking skeleton of aggregate particles, while the binder primarily acts as a lubricant to aid compaction and glue the particles together [49]. Generally, three techniques have been developed for characterizing the complex microstructure of asphalt mixture: the idealized model, the user-defined randomly created model, and the image-based model. In the idealized model, asphalt aggregates are represented by single-sized spherical particles randomly placed and mixed with the binder. In the randomly created model, properties of aggregates such as gradation, shape, angularity, orientation, and distribution are taken into consideration. The aggregates are randomly created and arranged to reach the desired microstructures of asphalt mixture. The image-based model is created by capturing the scanned images of an asphalt specimen, which is regarded as the most accurate model. Each of the three models has advantages over the others. Image-based models are usually lab-dependent and expensive, but can be regarded as a validating approach for a real and accurate calculation. Compared with the image-based model, the randomly created model is more efficient; it can represent different microstructures in the asphalt mixture, but it is not as accurate as the image-based model [50]. The simplified idealized model can be applied in preliminary studies for low cost. However, these imaging techniques need to coordinate with various numerical methods. DEM and FEM are two numerical techniques that are widely used to conduct micromechanical analyses of the heterogeneous microstructure of asphalt mixture [51], as shown in Fig. 3.

In the last few decades, DEM has been extensively applied to road engineering due to its good applicability for solving problems with non-continuous media and large deformation of pavement materials [52]. DEM was first proposed in the 1970s and originated from molecular dynamics [53]. It was initially developed for analyzing the mechanical properties of rock and soil from the mesoscopic aspect, and then extended to modeling solid materials, represented as an assembly of particles bonded in the interface. The translational and rational velocity of each particle is handled by all individual particles based on Newtonian mechanics as well as the appropriate inter-particle contacts [54]. The position of each particle is calculated through numerical integration. Though DEM is similar to molecular dynamics, it can be broadly distinguished by rotational degrees of freedom, stateful contacts, and complicated geometries. Different kinds of DEM computing codes have been developed, such as Trubal (a revised version of Ball), universal distinct element code (UDEC), discrete element code in three dimensions (3DEC), and particle flow code in two and three dimensions (PFC2D/3D) [55]. DEM is suitable for simulating and analyzing the granular properties of asphalt mixtures, and existing studies have provided meaningful insights into understanding the dynamic properties of asphalt mixtures using DE models [56, 57]. It is crucial to construct an accurate DE model for asphalt, and the correlation between mesoscale parameters and the macroscale material properties should be addressed [58]. Existing approaches for building a DE model fall into two categories: the image processing method and the computer generating method. For a typical DE model of asphalt mixture, coarse aggregates are characterized by irregular polyhedron particles, while the gaps are filled with voids and discrete elements of sand mastics [47]. After obtaining the 3D geometric model of aggregate particles, the 3D particle models are transformed into a cluster of sphere elements. The clusters of sphere elements are then assembled into the whole DE model of asphalt mixture with the appropriate contacts. Voids are generated by randomly deleting mastic elements according to the target concentration of air voids in the real samples. Finally, the DE model is imported into discrete element analysis software in order to conduct the simulation [59]. So far, the DE models used to represent the mechanical behaviors of asphalt mixture include the elastic model, viscoelastic model, and cohesive model [60]. These models have been developed and implemented for modeling the elastic, viscoelastic, and fracture behaviors of asphalt mixture. When taking time-dependent properties such as the viscoelasticity of asphalt binder into consideration, the difficulty of modeling increases. Burger’s constitutive relations have been input in order to represent the viscoelastic behaviors of asphalt mastic [61]. The viscoelastic properties relating to permanent deformation and rutting of asphalt mixture have been thoroughly studied in the last few decades [62,63,64]. For a better understanding of the fracture mechanisms of asphalt materials, an energy-based bilinear cohesive zone model has been implemented as a user-defined model within DEM to model the initiation and propagation of cracks [65]. Overall, the DE model is an excellent tool for investigating the influence of different characteristics of aggregates on the performance of asphalt mixtures. DE analyses indicate that the aggregate skeleton is the main bearing body under compressing force, and that it is critical for guaranteeing the overall stability of the aggregate skeleton and the bonding strength. However, it is almost impossible for DE simulation to take the fine particles fully into consideration, not only because this significantly increases computational time and cost, but also because it affects the system’s capability to reach equilibrium.

FE modeling provides accurate predictions for the physical properties of asphalt materials. Analyses of asphalt pavement by FEM were first made in 1968, and various research works were subsequently carried out to model the constitutive behaviors of asphalt mixtures. In the microstructure-based FE model of asphalt mixture, irregular aggregates and sand mastics are divided into different subdomains and meshes are generated within each aggregate and mastic subdomain. The aggregates and mastic elements share the same nodes on the aggregate boundaries for deformation connectivity [66]. The 2D microstructures of asphalt mixture can be obtained from the scanned image by optically scanning the smoothly sawn surface of compacted asphalt specimens, while the real 3D microstructure of three-phase asphalt mixture can be captured with X-ray CT [67, 68]. X-ray CT is an advanced technology for obtaining a series of sectional images of heterogeneous materials; it can provide comprehensively geometric information for numerical analyses such as grain topology, connectivity of air void, and behaviors of crack growth [69,70,71]. Compared to optical microscopes, X-ray CT can acquire accurate digital information from the internal microstructure and 3D geometry of solid material without damaging or destroying the samples. By taking time-lapse into consideration, X-ray tomography can be used to validate numerical predictions of structural deformation or microstructural evolution [72,73,74]. For example, the water transport inside asphalt mixture can be tracked by X-ray CT in order to analyze the mechanism of moisture damage [75]. After incorporating the nonlinear viscoelastic constitutive relation and geometric microstructural characteristics of aggregates and matrix, FE modeling can accurately predict the viscoelastic properties of asphalt mixture [76, 77].

Fracture behaviors are highly complicated, since they are essentially controlled by the complex morphological features and heterogeneous properties of asphalt mixture, which are dominated by the spatially random distributions of reinforced phase at mesoscale [78, 79]. During the mixing and compaction process, fillers coated with asphalt binder are suspended in the mix, close to the vicinity of larger aggregate particles. Following this, physical, chemical and mechanical actions may occur among the mastics and aggregates, leading to a narrow region forming around the aggregate particles. This region is called the interfacial transition zone (ITZ), which is usually the weak area, as it is the most prone to the generation of cracks [80, 81]. Traditional FEM is not appropriate for solving fracture problems in heterogeneous materials due to the large amount of re-meshing work. CZM has received increasing attention from pavement technologists and asphalt researchers over the past few decades, as it can model both brittle failure and ductile failure, which are commonly observed defects in asphalt materials due to the wide range of service temperatures and loading rates [82, 83]. CZM can predict the damage evolution in the fracture process zone located ahead of a crack tip. At the crack tip, the constitutive behavior of the cohesive zone reflects the change in the material properties due to the accumulation of damage ahead of the crack tip [84]. CZM is not limited to simulating a single crack tip, however, it can also describe the crack nucleation and widespread fractural phenomena at different time scales and length scales [85]. Cohesive elements are inserted at the boundary of solid elements along a pre-defined crack path, and a traction-separation law is used to characterize the behaviors of the interface of the fracture problems. The fracture mechanism is introduced by adopting a softening relationship (cohesive law) between the traction acting on the interface and the corresponding interfacial separation. The cohesive law in turn brings in cohesive fracture energy, which represents the energy required to break the interface surface [86]. This model involves nonlinear constitutive laws described by displacement jump and the corresponding traction along with the interfaces, and provides a proper phenomenological approach for simulating various fractural behaviors such as crack nucleation, crack initiation, and crack propagation [87]. CZM is applied to model fracture behaviors of cement mixture and asphalt mixture at mesoscale, which is also compatible with existing numerical methods such as FEM and DEM [78, 88].

Given the non-homogeneous characteristics of the asphalt mixture model, RVE can be adopted to convert nonuniform material into a corresponding equivalent homogeneous material for simplifying the calculation [89]. Homogenization theory has been developed in order to estimate the effective properties of a heterogeneous composite based on the microstructural description and local behaviors of its constituents. The RVE corresponds to a microstructural subdomain that represents the entire microstructure in an average sense, which contains adequate statistical information about the specific properties [90, 91]. The predicted stresses and strains within RVE can be spatially averaged, resulting in the constitutive behaviors of a homogenized material [92]. Theoretically, RVE represents a length-scale limit where the composite material can be modeled as a uniform continuum medium with homogenized properties, with the continuum mechanics and homogenization theories all based on the RVE concept. While RVE should certainly be smaller than the macroscopic body, it should also be large enough to contain sufficient details relating to the microstructure [93]. To minimize the computational cost, the optimum size of RVE is selected based on the results of the sensitivity analyses of sizes. The sizes of RVE are based on an assurance that the material can be assumed to be statistically homogeneous [94]. For granular composites, the minimum size of RVE is usually correlated to the maximum size of the particles [95]. Typically, the convergence of local fields in RVE requires a high-mesh density, and the computational resources are easily exhausted, which leads researchers to reduce their statistical requirements. In order to reduce the computational complexity, an inhomogeneous mesh density is used, which requires additional attention in order to obtain periodicity [96]. Though many effective techniques can be employed at macroscale and mesoscale to evaluate the characterization of asphalt materials, there are still many problems that cannot be totally understood and solved at smaller scales, such as the dynamical characteristics of molecular interactions [13, 97]. In other words, predictions of the properties of asphalt materials should be made from a fundamental bottom-up perspective.

## Molecular Modeling of Asphalt

The identification of asphalt components and the investigation of their respective properties have always been a key focus of asphalt study. The chemistry of asphalt molecules is still not well known, and over 95% of the asphalt molecules have not been isolated or identified. The light distillates with the lowest boiling points are widely known in detail, but the composition of heavier distillates is only known in general terms. Studies on the chemical structure help to develop the relationship between the nanostructure and the properties of asphalt. Asphalt consists of a large number of hydrocarbons with a small amount of sulfur, nitrogen, oxygen, and traces of metals such as vanadium and nickel [98]. These hydrocarbons are composed of polyaromatic structures or saturated polycyclic structures with different numbers of fused rings, saturated hydrocarbon side chains of various lengths, and substitution patterns. The element composition of asphalt depends primarily on its crude source, and it is difficult to give a geographical generalization and division of this [99]. In addition, the molecular structures of asphalt change during the refining process, and the properties of asphalt produced are obviously different from those that simply arise from the residue of crude oil distillation. The number of molecules with different chemical structures and molecular sizes is extremely large, and separation and identification of all the molecules would barely be possible. Furthermore, the number of possible isomers is almost unlimited, and none of them are in sufficiently large quantities to be isolated or characterized [100]. For decades, various computational modeling techniques at atomistic scale have been proposed for exploring the physicochemical and thermomechanical properties of asphalt. The first modeling technique applies the average molecular structure to represent different types of asphalt, and several technologies relating to the average models for whole asphalt structures have been found. According to experimental results, the average molecular weight of asphalt typically falls in the range of 600–1500 g/mol, and the distributions can vary depending on the experimental set-up [99]. Average molecular structures of eight core asphalts are built, and these are based on the results of solution-state NMR spectroscopy of asphalt composition, the distribution of molecular weight, and structure analyses. Several characteristics of asphalt are emphasized in this modeling technique, with the aromatic and aliphatic ratio, concentration of heteroatom, and types of functional groups. These standardized samples have been studied in the Strategic Highway Research Program (SHRP), which was a program launched by the US government in the 1990s aimed toward the mix designing method for new asphalt pavements [101]. In addition, a series of investigations have been carried out to construct the average molecular structure models of asphalt binders through analyses of element composition and structure parameters. Calibrations including GPC, FTIR, NMR, and elemental analyzer have been conducted based on the improved Brown-Ladner (B-L) methods, and the 3D micro-crack model has been further adapted to simulate the healing process of asphalt [102]. It has been found that average molecular structures can be useful in correlating the physical properties of real asphalt. However, the typical chemical properties of asphalt are based on its heterogeneity, particularly the unique combination of a large number of similar but different molecules, which makes it inefficient to describe asphalt using one or a few average-molecule models. Though these average models provide a convenient method for approximately simulating the properties of asphalt, the fraction and inner structures of the asphalt are ignored for adequate consideration.

From the molecular perspective, the complexity of asphalt compositions depends on the variables of both molecular sizes and the types of hydrocarbons. A lengthy and proper treatment is required in order to handle all these variables with complexity. After careful consideration, a simplified scheme that separates asphalt into different hydrocarbon types is emphasized [103]. The separation of asphalt components can be made using four classes or types, and each component is represented by a reasonably distinct generic hydrocarbon class. The functionality of asphalt molecules separates the classifications into different types that range in polarity, aromaticity, saturation, and size. The specific classes, for example, asphaltenes, polar aromatics (resin), naphthene aromatics, and saturates (SARA), can be physically isolated through selective adsorption-desorption and are sometimes instead classified as asphaltenes and maltenes, as shown in Fig. 4. Asphalt exhibits a colloidal structure that consists of asphaltenes surrounded by resins dispersed in saturates and aromatics [104]. The ratio of SARA can be significantly different depending on the type of asphalt. The ratio of SARA fractions is mainly influenced by various factors such as crude source, refining process, and oxidation extent [105].

Molecular dynamics (MD) simulation is a powerful modeling technology for elucidating various effects on mechanical performance and connecting mechanical properties and structural deformation to molecular interactions and motions in nanocomposite systems. Over the last decade, numerical efforts and achievements have been made using MD in the study of the physicochemical properties of asphalt materials [106, 107]. According to the solubility properties, asphalt is mostly modeled with three or four different components [108]. Asphaltene, saturates, and naphthene aromatics are used to build a three-component molecular model of asphalt [109,110,111]. The asphaltene structure is proposed to have moderate aromatic rings with small branches, and a group of molecular structures are set at a certain ratio to represent the asphaltene components in the asphalt model [109, 112]. The 1,7-dimethylnaphthalene is represented as the naphthene aromatics based on the ratio of alkane and aromatic, and the docosane is represented by the saturate in the asphalt model [113]. This model has been applied to study the physicochemical properties [110], moisture susceptibility [114], thermal expansion, and moduli analyses of asphalt [111], as well as the adhesion strength of the asphalt-aggregate interface. The agglomeration behaviors and adsorption mechanism of rubber-modified asphalt nanocomposites can also be studied by this model [115]. On the basis of the three-component asphalt model [109], some polar aromatic molecules are added in order to construct the six-molecule asphalt model. This shows that the six-molecule asphalt model has a very close elemental composition to that of real asphalt [116]. However, a more accurate and complex asphalt model is needed to characterize the behaviors of asphalt at the nanoscale.

With the four kinds of components, the twelve-molecule asphalt model is selected as the representative model for the well-known SARA frictions in virgin asphalt [117,118,119] by using Hansen solubility parameters, as shown in Fig. 5. Multiple molecules are proposed to represent the SARA fractions of asphalt binder in this model, including two molecules (squalane, hopane) for saturate, three molecules (asphaltene-phenol, asphaltene-pyrrole, asphaltene-thiophene) for asphaltene, five molecules (Quinolinohopane, thioisorenieratane, trimethylbenzeneoxane, pyridinohopane, benzobisbenzothiophene) for polar aromatics, and two molecules (PHPN, DOCHN) for naphthene aromatics. The molecular structures of polar aromatics accord with the findings of geochemistry, while the components of naphthene aromatics are molecules with slight molecular weight and low polarity. The mass percentages of the aromatic carbon and the aliphatic carbon and hydrogen of this asphalt model are calculated, and the density and concentration of different elements of this model accord with those in the experimental results. On the basis of this model, numerical studies have been carried out through MD to study the oxidation effect [119,120,121], interfacial adhesion [122,123,124], moisture effect [125], self-healing [126,127,128], rejuvenator effect [129], and micromechanical properties [130] of asphalt or asphalt nanocomposites. The twelve-molecule asphalt model has been further developed into the oxidized asphalt model [119]. It has been found through experiment that certain types of carbon and sulfur atoms of asphalt are susceptible to oxidation, and that ketones and sulfoxides are formed and identified as the main functional groups following oxidation. Therefore, the possible sites of pure asphalt models are replaced by the functional groups of ketones and sulfoxides to represent the oxidized asphalt model. Moreover, there are two developed oxidized asphalt models: one is the short-term oxidized model that simulates the oxidation during the construction process, and the other is the long-term oxidized model that represents the asphalt after years of service. The contents of the ketone and sulfoxide functional groups are adjusted according to the oxidized level of asphalt. Long-term oxidized asphalt has more ketone and sulfoxide functional groups than short-term asphalt, according to the FTIR results [121].

Four-component asphalt models have also been proposed, with each molecule represented within each solubility class. These four-component models can provide reasonable predictions for the physiochemical properties of asphalt such as viscosity and diffusivities dynamics [131], glass transition [132, 133], self-healing [134], and interactions [135]. An asphalt model with 20 molecules was chosen from the literature based on the different fractions [130]. and these 20 molecules cover most of the existing asphalt molecules. In addition, based on the measurements of GPC and FTIR, three-molecule models of different molecular sizes (small, medium, and large) have been proposed to represent asphalt models [104].

Although the full atomistic model can result in high accuracy and efficiency at the atomistic level, they still have limitations in terms of the length scale of nanometers and the time scale of nanoseconds. When considering molecular behaviors at larger scales, the coarse-grained (CG) model for asphalt is more powerful. A typical CG model for asphalt molecules is shown in Fig. 6. The major advantages of CG models are that they can save a lot more computational power than the atomistic model, without significantly lowering the accuracy. The basic principle of the CG model is to package a certain number of atoms into a single interaction group, which can simplify the complex molecular systems and unnecessary intramolecular interactions of MD simulations. The CG model of asphalt has been successfully adopted in investigating the diffusion and aggregation of asphalt molecules at the mesoscopic scale [136, 137].

## Forcefield and Application of MD

Forcefield plays a significant role in the accuracy of MD simulations. The validity of simulation results is related to the selection of forcefield or interatomic potential [138]. Many forcefields have been applied in studying the molecular behaviors of asphalt from the atomistic scale, such as COMPASS [139, 140], OPLS-AA [117], CHARMM [141], ACEFF [142], CVFF [143], PCFF [144], Dreiding [145], etc. In addition, the MARTINI forcefield has been adopted for the CGMD model of asphalt materials [137, 146]. Using these forcefields, the mechanical and physical properties of asphalt can be explored and understood. However, the forcefields are mostly selected from the empirical perspective and are not typically designed for asphalt. To bridge the gap between quantum chemistry and non-reactive empirical forcefields based on MD methods, ReaxFF has been designed to describe chemical reactions with bond formation and breaking. The application of Reactive MD simulations to the hydrocarbon system helps to provide information about the mechanism behind complicated reactive phenomena [147]. Furthermore, ReaxFF, as a bond-order-dependent forcefield, has been used to investigate the oxidation mechanism and the self-healing behaviors of asphalt [148]. Experimental measurements have verified the results of Reactive MD simulations, and the obtained asphalt model is also considered valid [149].

There are different kinds of molecular interactions inside asphalt. The first is the London dispersive interaction, which is generated by the temporary dipoles of non-polar hydrocarbon molecules. Another non-polar interaction is π–π stacking, which is caused by the attractive aromatic rings of hydrocarbons [150]. Polar interactions such as H-bond interactions exist among the electronegative elements of polar hydrocarbons, such as sulfur, nitrogen, and oxygen. Since the polar elements only constitute a small amount of the asphalt, the dispersive interaction dominates in the molecular interactions. The interfacial adhesion plays a vital role in maintaining the integrity and the overall performance of asphalt materials. The intermolecular interactions at the wet asphalt/aggregate interface are more complicated than those of the bulk asphalt, because of the asphalt/mineral and water/mineral interactions, as well as the fact that the interface between asphalt and aggregate in asphalt mixture is mainly focused on the adhesion and cohesion performance of asphalt and aggregate [151, 152]. In comparison with bulk asphalt, the adhesive properties of the asphalt/aggregate interfacial transition zone are more susceptible to moisture, and the interfacial debonding proceeds to strip. It is still a big challenge to elucidate the failure mechanisms that occur during exposure to the moisture environment because of the lack of sufficient information on the nanostructure of the asphalt/aggregate interface [153]. Four representative minerals – quartz, calcite, albite, and microcline–have been selected to investigate adhesion properties and debonding behaviors with asphalt through MD. It has been found that the adhesion between minerals and asphalt can be attributed to non-bond interaction energy, where the major contribution is van der Waals interaction for neutral minerals and electrostatic interaction for the alkali minerals [154]. The micro-mechanisms of interfacial adhesion failure between nano-silica modified asphalt and aggregates have been investigated, and the effects of nano-silica, aggregate surface irregularity, and seawater erosion on the interface have been considered by MD simulations [155]. The surface free energy measurement and molecular dynamic simulation can potentially provide alternative quantitative methods for the adhesion evaluation from raw materials to structures [156]. Based on the references mentioned above, the feasibility and validity of MD modeling techniques for asphalt materials can be demonstrated. Moreover, MD simulation is helpful for uncovering the interactive details of asphalt molecules, which cannot be obtained from the bulk measurement. The nanoscale properties of the asphalt binder significantly affect the macroscale properties due to the fast development of technology. As a supercomputer with unpredicted computational power may be accessible in the future, the model size and the time scale will obviously increase. As such, the simulation length of the MD simulations will be closer to that of the macro test, and the simulation result will be more representative of the test results of asphalt.

Despite the importance of a molecular-level understanding of asphalt’s macroscopic properties, the complexity and massive size of asphalt structural units, along with the limited available computational power, have reduced theoretical approaches mainly to the classical MD simulations. There are very few preliminary quantum-based studies at the density functional theory (DFT) level that provide a proper description of electronic effects. The mechanism of aged asphalt and the intermolecular interaction of bio-oil/asphalt composite at the molecular and electronic levels are analyzed using DFT models [157, 158].

## Multiscale Modeling

Continuum approaches do not explicitly model the granular structure of asphalt mixture, but treat the composite in a smear fashion. Mesoscale modeling explicitly takes the grains into account via geometric representation. Multiscale models can bridge the gap between mesoscale and continuum approaches by performing homogenization at the smaller scales in order to obtain the smeared parameters for use at the continuum scale [77]. The multiscale modeling approaches allow the analyses of materials and structural behaviors to be performed separately at multiple length scales and link them through a homogenization process carried out within RVE. In light of this advantage, multiscale modeling provides an efficient means of accounting for the inelasticity, heterogeneity, and damage/fracture of pavement materials and structures, as well as their temporal and spatial evolution under external environmental conditions [159]. Some researchers have applied microstructure-based multiscale modeling to rutting and bottom-up cracking analyses for asphalt pavements, while several studies have focused on the top-down cracking distress of asphalt pavements [160]. The underlying mechanisms between wax and asphalt have been examined by multiscale experiments and MD simulations. It has been found that the presence of wax leads to improved workability of the asphalt mixture and lowers the mixing and compaction temperature of asphalt [161].

The self-healing behaviors of asphalt materials have been explored using the multiscale modeling method. At macroscale, asphalt is characterized by several indexes such as penetration grade, softening point, viscosity, and moduli. It has been found that asphalt with a higher penetration grade and lower softening point has a higher self-healing capability, since the asphalt can flow into the crack areas more easily with the effect of capillary force. At the mesoscale, self-healing is considered a cohesive behavior within the asphalt mastic or adhesive behaviors when it happens at the asphalt/aggregate interface. At the nanoscale, self-healing behaviors are related to molecular mobility, chemical composition, atomistic structures, and temperature ranges [162]. Nevertheless, there are no agreed criteria for evaluating the self-healing mechanism of asphalt on a single scale, and a comprehensive understanding of the self-healing behaviors of asphalt is needed. Besides, multiscale modeling for DFT and MD has been applied to interpret the synergistic and antagonistic interactions between asphalt and modifiers, including polyphosphoric acid (PPA) and nano-silica [163]. The DFT results and energy decomposition analyses show that the silica surface forms an efficient interaction with PPA through noncovalent interactions, increasing the PPA-asphalt interactions.

## Prospects and Challenges

With the development of smart cities, research studies on intelligent asphalt pavements have become the focus of urban development. Traditional pavements will be replaced gradually by intelligent ones, and asphalt pavement with multi-functionality and high durability will become the norm. Damaged asphalt will be able to obtain this self-healing ability and restore its performance automatically. Asphalt pavement will become more durable and able to deal with different facets of drastic climate change and heavy traffic volume. Technologies with new reinforcing materials and advanced construction will help to improve the performance of asphalt materials. With the development of manufactural approaches, nanomaterials have become cheaper and more popular when modifying asphalt. Novel modeling methods for asphalt nanocomposites will be addressed in order to uncover the reinforcing mechanism of nanomaterials and the interactions between asphalt and nanomaterials. Modifiers with environmental and friendly attributes will become more popular in the modification of asphalt, and reclaimed asphalt pavement with better recyclability will save a significant amount of energy and cost in the construction of new asphalt pavements. Moreover, the modeling technology for asphalt will become more significant. Multiscale modeling has the advantages of quantitatively calculating the performance of asphalt materials at the macroscale and qualitatively analyzing the mechanism of different components at the microscale. With the development of computational power, MD simulations will play a critical role in modeling asphalt materials. The modeling methods at the atomistic scale will potentially bridge the properties of asphalt at macroscale, which will provide a bottom-up perspective for engineers and researchers in the pavement field.

## Conclusions

In this paper, the development and prospects of asphalt materials modeling, including modeling at macroscale, mesoscale, and nanoscale, have been reviewed. The continuum model at the macroscale has a high efficiency in calculating the mechanical properties of asphalt mixture and predicting the practical performance of asphalt pavement. However, the macroscopic model cannot elucidate the roles of different constituents of asphalt mixture in the degradation process. The microstructural models at the mesoscale consist of aggregates, air voids, and asphalt mastic, which are more complex but also more accurate than the continuum models at macroscale. The types and shapes of aggregates, the distributions of air voids, and the physicochemical properties of asphalt mastic can directly determine the mechanical properties of asphalt mixture, and there are three main approaches for building microstructural models, namely the idealized model, the randomly created model, and the image-based model. Among them, the image-based model is the most accurate model, since it is based on data from real asphalt mixture. Due to the heterogeneous nature of the microstructure model, numerical methods such as DEM and FEM have been adopted in order to analyze its micromechanical properties with high accuracy. CZM is especially able to handle the fracture problems of microstructural models; it has better efficiency than FEM, which needs significant re-meshing work at the crack zone.

With the deeper and further study of asphalt, it has been realized that asphalt materials are rather complex, and that the asphalt binder is not homogeneous at nanoscale, in contrast to macroscale and mesoscale. Various experimental techniques have been applied to explore the structures of asphalt at nanoscale, such as FTIR, NMR, GPC, and various microscopes. According to the chemical properties of asphalt, two classes of asphalt models at the nanoscale–the average-molecule model and multi-molecule model–are proposed to represent the molecular structures of asphalt. The average model mixes average molecules to characterize the physical properties of asphalt, and is a simplified approach to simulating its properties. However, the complexity of different compositions of asphalt cannot be fully characterized. Asphalt consists of varying hydrocarbon molecules and non-hydrocarbon molecules with different shapes, sizes, and polarities. Based on asphalt chemistry, there are four main components of asphalt: asphaltene, polar aromatics, naphthene aromatics, and saturates. Each of these components can be represented by one or several molecules, and the twelve-molecule asphalt model has been proposed and widely adopted in asphalt research for mechanical interactions, oxidation, moisture, etc. In addition, asphalt models including three-molecule, four-molecule, six-molecule and twenty-molecule have been tried and applied when solving different problems at the molecular level. To extend the application of MD simulations on the length scale, CG models have been adopted at the mesoscale level, as they require much less computational power than full atomistic models.

Multiscale modeling has become the trend in modeling asphalt materials due to the development of computation power and modeling technology. It can provide a more comprehensive insight from the multiscale perspective than single scale modeling, since modeling from one scale has its own limitations in terms of efficiency and accuracy. A typical multiscale approach is RVE, which can convey structural information at the mesoscale to the continuum modeling of macroscale. Multiscale modeling for DFT and MD modeling can bridge atomistic behaviors with the evolution of electronic structure. Overall, multiscale modeling can break through the obstacles of asphalt modeling on one scale and provide the advantages of both efficiency and accuracy at different scales.

The prospects and future of asphalt pavements are such that they will be more intelligent than traditional versions. The intelligent pavement will be able to provide multifunctional properties like self-healing, self-cleaning, deicing, etc. This improved functionality will demand better technology for modeling asphalt materials. There are several challenges in the modeling work of asphalt. One is searching for and creating a highly accurate atomistic model of asphalt, which is still absent at this stage. Another is developing a specific forcefield for asphalt-based materials, since most of the adopted forcefields at this stage are based on polymeric materials. It is believed that with the development of modeling methods for asphalt, the failure mechanisms of asphalt pavement will be further understood and novel reinforcement methods will be able to obtain high durability.

## Data availability

No customized code or software program was used in this review study.

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## Acknowledgements

The authors are grateful for the support from the Research Grants Council (RGC) of the Hong Kong Special Administrative Region, China (Grant No. CityU11209418).

## Funding

This work was supported by Research Grants Council, University Grants Committee (Grant No. CityU11209418).

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FN: Collected the literature and drafted the manuscript. CLC: Reviewed and edited the manuscript. DL: Supervised and reviewed the manuscript.

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### Cite this article

Nie, F., Chow, C.L. & Lau, D. A Review on Multiscale Modeling of Asphalt: Development and Applications.
*Multiscale Sci. Eng.* **4**, 10–27 (2022). https://doi.org/10.1007/s42493-022-00076-x

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DOI: https://doi.org/10.1007/s42493-022-00076-x

### Keywords

- Multiscale
- Asphalt
- Modeling technique
- Molecular dynamics simulation
- Durability